Transfer-closed characteristicity is transitive

This article gives the statement, and possibly proof, of a subgroup property (i.e., transfer-closed characteristic subgroup) satisfying a subgroup metaproperty (i.e., transitive subgroup property)
View all subgroup metaproperty satisfactions | View all subgroup metaproperty dissatisfactions |Get help on looking up metaproperty (dis)satisfactions for subgroup properties
Get more facts about transfer-closed characteristic subgroup |Get facts that use property satisfaction of transfer-closed characteristic subgroup | Get facts that use property satisfaction of transfer-closed characteristic subgroup|Get more facts about transitive subgroup property

Definition

Suppose $H\leq K\leq G$ are groups such that $K$ is a transfer-closed characteristic subgroup of $G$ and $H$ is a transfer-closed characteristic subgroup of $K$. Then, $H$ is a transfer-closed characteristic subgroup of $G$.

Facts used

1. Characteristicity is transitive
2. Transfer condition operator preserves transitivity

Proof

Proof using given facts

The proof follows from facts (1) and (2).

Hands-on proof

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